Descent for projective twists of modular curves

Let F be a number field and p≥7 a rational prime. We obtain a simple descent criterion characterising those projective Galois representations ρ:GF₂(Fₚ) for which the corresponding twist Xρ(p) of the principal modular curve of level p is defined over Q. We also give a more concrete version of this result for representations which arise from elliptic curves over cyclic number fields.